Electrical engg material

Unit 1
Electrical Theory

AC
• Battery symbol is used as a
generic symbol for any DC
voltage source
• The circle with the wavy line
inside is the generic symbol
for any AC voltage source

Generation of AC
• Simpler Design
• Naturally
Produced
• No brushes for
Motor
• Greater Reliability
• Low Cost

Generation of DC
• Complex Design
• Commutator
• Sparks & Heat
• DC Motor Brushes
• Dearer

AC Waveform
Angle (o
) sin(angle) wave Angle (o
) sin(angle) wave
0 0.0000 zero 180 0.0000 zero
15 0.2588 + 195 -0.2588 -
30 0.5000 + 210 -0.5000 -
45 0.7071 + 225 -0.7071 -
60 0.8660 + 240 -0.8660 -
75 0.9659 + 255 -0.9659 -
90 1.0000 +peak 270 -1.0000 -peak
105 0.9659 + 285 -0.9659 -
120 0.8660 + 300 -0.8660 -
135 0.7071 + 315 -0.7071 -
150 0.5000 + 330 -0.5000 -
165 0.2588 + 345 -0.2588 -
180 0.0000 zero 360 0.0000 zero

RMS?????
• An AC measurement based on work performed by a
waveform is not the same as that waveform's “average” value
• The power dissipated by a given load (work performed per
unit time) is not directly proportional to the magnitude of either
the voltage or current impressed upon it
• Power is proportional to the square of the voltage or current
applied to a resistance (P = E2/R, and P = I2R)
• Method of deriving an aggregate value for waveform
amplitude is based on the waveform's ability to do useful work
when applied to a load resistance

Bandsaw Analogy (DC vs AC)

VRMS
• The qualifier “RMS” stands for Root Mean Square
• The procedure consists of squaring all the positive and negative
points on a waveform graph, averaging those squared values, then
taking the square root of that average to obtain the final answer
• RMS value of an alternating waveform is the value that, when
applied across a resistance, produces that same amount of heat
that a direct current (DC) voltage of the same magnitude would
produce
• RMS amplitude (VRMS) measurement is the best way to relate AC
quantities to DC quantities, or other AC quantities of differing
waveform shapes, when dealing with measurements of electric
power
• 𝑉𝑃𝐾 = 2 . 𝑉𝑅𝑀𝑆

VRMS, VPK, VPK-PK, VAV
• VRMS is the Root Mean Square Value of Alternating Current. Also sometimes termed
as DC Equivalent.
• VPK is a way to express the intensity, or magnitude (also called the amplitude), of an
AC quantity to measure its peak height on a waveform graph
• VPK-PK is another way to measure the total height between opposite peaks
• VAV is the mathematical average of the values of all the points on a waveform's graph
to a single, aggregate number.
• When determining the proper size of wire (ampacity) to conduct electric power from
a source to a load, RMS current (IRMS) measurement is the best to use, because the
principal concern with current is overheating of the wire
• When rating insulators for service in high-voltage AC applications, peak voltage
(VPK) measurements are the most appropriate, because the principal concern here is
insulator “flashover” caused by brief spikes of voltage, irrespective of time

Derive VRMS
𝑉𝑅𝑀𝑆 =
1
𝜋 2 0
𝜋
2
𝑉𝑃𝐾 sin 𝜃 2 . 𝜕𝜃
𝑉𝑅𝑀𝑆 =
2. 𝑉𝑃𝐾
2
𝜋
𝜃
2
−
1
4
sin 2 − 𝜃 0
𝜋/2
𝑉𝑅𝑀𝑆 =
2 𝑉𝑃𝐾
√𝜋
𝜋 2
2
−
1
4
sin 𝜋 −
0
2
−
1
4
sin 0
𝑉𝑅𝑀𝑆 =
2𝑉𝑃𝐾
𝜋
𝜋
4
− 0 − 0 − 0
𝑉𝑅𝑀𝑆 =
2𝑉𝑃𝐾
𝜋
.
𝜋
2
=
1
2
𝑉𝑃𝐾
VRMS =
1
2
. 𝑉𝑃𝐾
VRMS = 0.707 . 𝑉𝑃𝐾

Derive VAV
𝑉𝐴𝑉 =
1
𝜋 2 0
𝜋 2
𝑉𝑃𝐾 . sin 𝜃 . 𝛿𝜃
𝑉𝐴𝑉 =
𝜋
2
. 𝑉𝑃𝐾 − cos 𝜃 |𝜋 2
0
𝑉𝐴𝑉 =
−2
𝜋
. 𝑉𝑃𝐾 cos
𝜋
2
− cos 0
𝑉𝐴𝑉 =
−2
𝜋
. 𝑉𝑃𝐾 0 − 1
𝑉𝐴𝑉 =
2
𝜋
. 𝑉𝑃𝐾
𝑉𝐴𝑉 = 0.636 . 𝑉𝑃𝐾

Proof of Derivation
Angle (o) sin(angle) wave Angle (o) sin(angle) wave
0 0.0000 zero 180 0.0000 zero
15 0.2588 + 195 -0.2588 -
30 0.5000 + 210 -0.5000 -
45 0.7071 + 225 -0.7071 -
60 0.8660 + 240 -0.8660 -
75 0.9659 + 255 -0.9659 -
90 1.0000 +peak 270 -1.0000 -peak
105 0.9659 + 285 -0.9659 -
120 0.8660 + 300 -0.8660 -
135 0.7071 + 315 -0.7071 -
150 0.5000 + 330 -0.5000 -
165 0.2588 + 345 -0.2588 -
180 0.0000 zero 360 0.0000 zero
Description
Full
Cycle
Qtr Cycle
Sum 1719.1 429.8
Count 28.0 7.0
Mean (Avg
Voltage)
61.4 61.4
Sum of Squares 139995.5 34998.9
Mean of Squares 4999.8 4999.8
Root of Mean of
Squares
70.7 70.7

Additional Information
• The conversion constants shown here for peak, RMS, and average
amplitudes of sine waves, square waves, and triangle waves hold
true only for pure forms of these wave shapes
• The crest factor of an AC waveform, is the ratio of its peak (crest)
value divided by its RMS value
• The form factor of an AC waveform is the ratio of its RMS value
divided by its average value
• The amplitude of an AC waveform is its height as depicted on a
graph over time

Three Phase Advantages
• The horsepower rating of three phase motors and the KVA
rating of three phase transformers are 150% greater than
single phase motors or transformers of similar frame size
• The power delivered by a single phase system pulsates and
falls to zero. The three phase power never falls to zero. The
power delivered to three phase system is the same at any
instant. This produces superior characteristics for three-phase
motors
• A three phase system needs three conductors, however each
conductor is only 75% the size of equivalent KVA rated single
phase conductors

Single Phase Systems
P1 = 20 kW P2 = 20 kW
Load 1 Load 2
220 V AC
• 𝐼 =
𝑃
𝐸
=
20000
220
= 90.9 𝐴𝑚𝑝𝑠
• 𝐼𝑡𝑜𝑡𝑎𝑙 = 𝐼𝐿1 +
𝐼𝐿2 = 90.9 + 90.9 = 181.8 𝐴𝑚𝑝𝑠
• Ptotal = P1 + P2 = 20 kW + 20 kW = 40kW
• Copper Size???
• Copper Weight???
• Copper Cost???

Single Phase Systems
Load 1 Load 2
P1 = 20 kW P2 = 20 kW
440 V AC
• 𝐼 =
𝑃
𝐸
=
20000
440
= 45.45 𝐴𝑚𝑝𝑠
• 𝐼𝑡𝑜𝑡𝑎𝑙 = 𝐼𝐿1 +
𝐼𝐿2 = 45.45 + 45.45 = 90.9 𝐴𝑚𝑝𝑠
• Ptotal = P1 + P2 = 20 kW + 20 kW = 40kW
• Same power at half the cost, size and weight

Modified Power System
• Current is sill same
• Power is same
• What if one of the
LOAD is open?
• If either load is open,
current will stop
Load 1
Load 2
220 V
20 kW
220 V
20 kW
440 V AC
45.45 A
45.45 A
440 V

Split-Phase System
• More efficient
• More Safe
Load 1
Load 2
220 V
20 kW
220 V
20 kW
220 V AC
45.45 A
45.45 A
440 V
220 V AC
0 A

TRIGONOMETRY FUNCTION
θ 𝟎° 𝟑𝟎° 𝟒𝟓° 𝟔𝟎° 𝟗𝟎°
sin 𝜃 0
1
2
1
2
3
2
1
cos 𝜃 1
3
2
1
2
1
2
0
θ 𝟗𝟎° 𝟏𝟐𝟎° 𝟏𝟑𝟓° 𝟏𝟓𝟎° 𝟏𝟖𝟎°
sin 𝜃 1
3
2
1
2
1
2
0
cos 𝜃 0 −
1
2
−
1
2
−
3
2
− 1

Complex Numbers
• Z = x + jy
• Z - is the Complex
Number representing the
Vector
• x - is the Real part or the
Active component
• y - is the Imaginary part
or the Reactive component
• j - is defined by −1

POLAR FORM
• Z = A ∠θ
• 𝐴2 = 𝑥2 + 𝑦2
• 𝐴 = 𝑥2 + 𝑦2
• 𝑥 = 𝐴 cos 𝜃
• 𝑦 = 𝐴 sin 𝜃
• 𝜃 = tan−1 𝑦
𝑥

Converting Polar to Complex
• Given 6 ∠ 30 o
• 6 ∠ 30 o= 𝑥 + 𝑗𝑦
• 𝑥 = 𝐴 cos 𝜃, 𝑦 = 𝐴 sin 𝜃
• 6 ∠ 30 o= 6 cos 𝜃 + 𝑗6 sin 𝜃
• 6 ∠ 30 o= 6 cos 30° + 𝑗6 sin 30°
• 6 ∠ 30 o= (6 x 0.866) + j (6 x 0.5)
• 6 ∠ 30 o= 5.2+ j3

Converting Complex to Polar
• Given 5.2+ j3
• 5.2+ j3 = A ∠θ
• A = 5.22 + 32
• A = 6
• 𝜃 = tan−1 𝑦
𝑥
• 𝜃 = tan−1 3
5.2
= 30 O
• 5.2+ j3 = 6 ∠30 o

Three Phase Power System
• 220 ∠ 0 o - 220 ∠ 120 o
• 220 + 𝑗0 + 110 + 𝑗190
• 330 + j190
• 380 ∠ -30 o
Load 1
Load 2
220 V
20 kW
220 V
20 kW
220 V AC ∠ 120 o
45.45 A
45.45 A
380 V
220 V AC ∠ 0 o

• The 3rd wire carries
current
• Therefore, we could have
a voltage source on the
3rd wire also
• Thus achieving a 3 phase
power systems
Load 1
Load 2
220 V
20 kW
220 V
20 kW
220 V AC ∠ 120 o
45.45 A
45.45 A
440 V
0 A
220 V AC ∠ 240 o
220 V AC ∠ 0 o

• Three-Phase system is the most common method used by electric
power distribution grids worldwide to distribute power
• A three-phase system is generally more economical than others
• Three-phase system was introduced and patented by Nikola Tesla

• The phase currents tend to cancel out one another, summing to
zero in the case of a linear balanced load.
• This makes it possible to eliminate or reduce the size of the neutral
conductor;
• all the phase conductors carry the same current and so can be the
same size, for a balanced load.
• Power transfer into a linear balanced load is constant, which helps
to reduce generator and motor vibrations.
• Three-phase systems can produce a magnetic field that rotates in a
specified direction, which simplifies the design of electric motors.

Y and Δ Connection
STAR(Υ) DELTA (Δ)
IL = IPH IL = 3. IPH
VL = 3. VPH
VL = VPH
TOTAL POWER = 3. VL.IL TOTAL POWER = 3. VL.IL

Where Does that 3 Come From
• Va = VPH ∠ 0 o; Vb = VPH ∠ -120 o; Vc = VPH ∠ -240 o
• Vab = Va - Vb
• VL = VPH ∠ 0 o - VPH ∠ -120 o
• VL = VPH (1 - 1 ∠ -120 o)
• VL = VPH {1 - (cos 120 o - jsin 120 o)}
• VL = VPH {1 - [(-
1
2
)- j(
3
2
)]}
• VL = VPH {1 - [(-
1
2
)+j (
3
2
)]}
• VL = VPH {
3
2
+j
3
2
} = VPH
𝟑
𝟐
+ VPH
𝟑
𝟐

Where Does that 3 Come From
• VL = VPH
3
2
2
+
3
2
2
∠ tan−1
( 3 2
3 2)
• VL = VPH
9
4
+
3
4
∠ tan−1( 1 3)
• VL = 𝟑 . VPH ∠ 30 o
• VL = 𝟑 . VPH
• Similary, you may solve for IL and IPH

Terms and Naming Conventions
• Phase: Describes or pertains to one element or device in a load, line, or source. It is simply a "branch" of the
circuit and could look something like this .
• Line : Refers to the "transmission line" or wires that connect the source (supply) to the load. It may be modeled as
a small impedance (actually 3 of them), or even by just a connecting line.
• Neutral: The 4th wire in the 3-phase system. It's where the phases of a Y connection come together.
• Phase Voltages & Phase Currents: The voltages and currents across and through a single branch (phase) of the
circuit. Note this definition depends on whether the connection is Wye or Delta!
• Line Currents :The currents flowing in each of the lines (Ia, Ib, and Ic). This definition does not change with
connection type.
• Line Voltages :The voltages between any two of the lines (Vab, Vbc, and Vca). These may also be referred to as
the line-to-line voltages. This definition does not change with connection type.
• Line to Neutral Voltage: The voltages between any lines and the neutral point (Va, Vb, and Vc). This definition
does not change with connection type, but they may not be physically measureable in a Delta circuit.
• Line to Neutral Current: Same as the line currents (Ia, Ib, and Ic).

Thank you!
company presentation 2012
27/09/2021

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